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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Variance-gamma distribution ： ウィキペディア英語版 Variance-gamma distribution \; e^ $K\_\backslash lambda$ denotes a modified Bessel function of the second kind $\backslash Gamma$ denotes the Gamma function| cdf =| mean =$\backslash mu\; +\; 2\; \backslash beta\; \backslash lambda/\; \backslash gamma^2$| median =| mode =| variance =$2\backslash lambda(1\; +\; 2\; \backslash beta^2/\backslash gamma^2)/\backslash gamma^2$| skewness =|| kurtosis =|| entropy =| mgf =$e^\; \backslash left(\backslash gamma/\backslash sqrt\backslash right)^$| char =| }} The variance-gamma distribution, generalized Laplace distribution or Bessel function distribution〔 is a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the gamma distribution. The tails of the distribution decrease more slowly than the normal distribution. It is therefore suitable to model phenomena where numerically large values are more probable than is the case for the normal distribution. Examples are returns from financial assets and turbulent wind speeds. The distribution was introduced in the financial literature by Madan and Seneta.〔D.B. Madan and E. Seneta (1990): The variance gamma (V.G.) model for share market returns, ''Journal of Business'', 63, pp. 511–524.〕 The variance-gamma distributions form a subclass of the generalised hyperbolic distributions. The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available. The class of variance-gamma distributions is closed under convolution in the following sense. If $X\_1$ and $X\_2$ are independent random variables that are variance-gamma distributed with the same values of the parameters $\backslash alpha$ and $\backslash beta$, but possibly different values of the other parameters, $\backslash lambda\_1$, $\backslash mu\_1$ and $\backslash lambda\_2,$ $\backslash mu\_2$, respectively, then $X\_1\; +\; X\_2$ is variance-gamma distributed with parameters $\backslash alpha$, $\backslash beta$, $\backslash lambda\_1+\backslash lambda\_2$ and $\backslash mu\_1\; +\; \backslash mu\_2$. The variance-gamma distribution can also be expressed in terms of three inputs parameters (C,G,M) denoted after the initials of its founders. If the "C", $\backslash lambda$ here, parameter is integer then the distribution has a closed form 2-EPT distribution. See 2-EPT Probability Density Function. Under this restriction closed form option prices can be derived. See also Variance gamma process. == Differential equation == The pdf of the variance-gamma distribution is a solution of the following differential equation for $x>u$: :$\backslash left\backslash$ (x-\mu ) f''(x)-2 f'(x) (-\beta\mu+\lambda+\beta x-1)+ f(x) \left(\alpha^2 \mu-\beta (\beta\mu-2 \lambda+2)+ x \left(\beta^2-\alpha^2\right)\right)=0, \\() f(0)=\frac\right)^} e^ \mu^} \left(\alpha-\frac\right)^ K_}(-\alpha\mu)}, \\() f'(0)=\frac-\lambda} \mu e^ (-\mu )^} \left(\alpha-\frac\right)^ \left((\beta\mu-2 \lambda+1) K_}(-\alpha\mu)- \alpha\mu K_}(-\alpha\mu)\right)} \end\right\} It is a solution of the following differential equation for :$\backslash left\backslash$ (x-\mu ) f''(x)-2 f'(x) (-\beta\mu+\lambda+\beta x-1)+ f(x) \left(\alpha^2 \mu-\beta(\beta\mu-2 \lambda+2)+ x \left(\beta^2-\alpha^2\right)\right)=0, \\() f(0)=\frac-\lambda}\sqrt} e^ \left(\mu \left(\alpha- \frac\right)\right)^ K_}(\alpha\mu)}, \\() f'(0)=\frac-\lambda} e^ \mu^} \left(\alpha-\frac\right)^ \left((\beta\mu-2 \lambda+1) K_}(\alpha\mu)+\alpha\mu K_}(\alpha\mu)\right)} \end\right\} 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Variance-gamma distribution」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.